Find the slope of the secant line PQ for the following values of x?

Question

Given that Point P (1, 1/2) lies on the curve y = x / (1+x) and Q is a point (x, x/(1+x))
i. 0.5
ii. 09.9
iii. 0.99
iv. 0.999
v. 1.5
vi. 1.1
vii. 1.01
viii. 1.001

So I wrote the equation M ={ x/(1+x) - (1/2) } / (x-1) and put it into the calculator along with y = x/1+x. When I go to table and put in the values for x I get the wrong answers for everything except 0.5. What am I doing wrong? Thanks a lot.

Yes, those are the correct answers. I can get them if I manually plug them into my equation that I listed above. I'm trying to do this on a T-84

y1= x/(1+x)
y2 = x/(x+1)-(1/2)/(x-1)
Then I hit 2nd table and enter the value 0.9 and just get the answer to plugging that number into the first equation as Y1 and the wrong answer as Y2. How can I make the calculator give me the slope of the secant?

Answer 1

clearly you're sneaking up on the tangent at x = 1 from below and above (and ii. should be 0.9):

x ............dy/dx
.5 .......... 0.33333333333
.9 .......... 0.26315789
.99 ........ 0.25125628
.999 ...... 0.25012506
1.5 ........ 0.2
1.1 ........ 0.23809524
1.01 ...... 0.24875622
1.001 .... 0.24987506

So it sure looks like the limit is 1/4. If you're not getting this, probability is 99% that you made a mistake with the parentheses.